Green's theorem

English

Etymology

Named after the mathematician George Green.

Noun

Green's theorem (uncountable)

1. (calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

   $\iint _{R}\left({\partial Q \over \partial x}-{\partial P \over \partial y}\right)dx\,dy=\oint _{\partial R}P\,dx+Q\,dy$.

2. (calculus) Letting ${\vec {G}}=(P,Q)$ be a vector field, and $d{\vec {l}}=(dx,dy)$ this can be restated as

   $\iint _{R}\nabla \wedge {\vec {G}}\ dx\,dy=\oint _{\partial R}{\vec {G}}\cdot d{\vec {l}}$

   where $\wedge$ is the wedge product, or equivalently, as

   $\iint _{R}\nabla \cdot {\vec {G}}\ dx\,dy=\oint _{\partial R}{\vec {G}}\wedge d{\vec {l}}$,

   with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.